Does the metal bracket offer more hole locations than you actually need? If so, perhaps you could just drill a new hole...
– unutbuJan 20 at 12:33

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@unutbu, hole location at the top are not possible as the wall structure is weak. So the better option is to place the screw at the side
– hunterexJan 20 at 12:40

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Can you edit your question to say what the wall is made of please?
– Martin BonnerJan 21 at 9:32

@MartinBonner - indeed. I think far to many people are assuming everybody uses plasterboard/drywall everywhere... that wall looks far more like there's going to be brick or block behind it; in which case the answer would be far different [double-depth hole & longer screw to reach a rawl plug in the harder substrate]
– TetsujinJan 23 at 11:44

Also, in general, it's important to check the load ratings of anchors. In OP's case we can see the bracket and that limits the screw size (and anchor rating) so it's likely not an issue, but some blinds can be heavy and need larger anchors if going into drywall and not into a stud.
– J...Jan 21 at 14:14

An old trick is to put tooth picks or matches (without the combustible material) into the hole with some glue to fill it and the put put the screw in. But the plugs suggested is a much better solution.
– nivs1978Jan 22 at 13:24

2

These aren't really designed for plaster and will often crack the plaster when you put in the screw. One trick I found was to drill an even bigger hold in the plaster and then 'glue' an anchor in with joint compound for light duty needs.
– JimmyJamesJan 22 at 15:20

@JimmyJames Excellent point. Drywall anchors are really designed for drywall - they need the paper cladding to hold everything together. It's not clear if OP's wall is plaster/lath or drywall, but it looks like it could be either. If it's not drywall then these are definitely not suitable.
– J...Jan 23 at 12:58

@J... Yeah I'm not totally clear on what 'plaster board' is exactly. It could just be drywall. I had a plumbing access cover literally just drop off the wall one day because it was secured with screws into plaster which had cracked. Toggle bolts wouldn't work so I made holes larger than the drywall anchors and pushed them in with joint compound. Good enough to get the panel back in place but I would not use this technique for anything weight bearing.
– JimmyJamesJan 23 at 15:47

If there is no wood at all behind this spot anywhere, a plain wall anchor may rip out over time - curtain rails hold curtains which can be heavy, and can flap around in the wind creating vibration which slowly erodes plaster.

Rightmost is your standard cheap wall insert. It is supposed to flange out as the screw goes through the middle. Avoid where possible.

Second from right is a plastic toggle. These are pretty good because the two bits on the side fold and press against the inside of the plaster board. Ideally they should be sized for the thickness of your board.

Middle shows the large-radius coarse thread variant of the first one. These are astonishingly good. For double the radius you're getting at least four times the holding power.

Second from left is the toggle bolt. It needs a slightly bigger hole for the legs to go through, but when expanded inside the wall (as pictured, its got a spring) then there's a heap of holding area. Can hold medium sized mirrors.

Leftmost is a "mollybolt" and is an older design, harder to find. This one has four legs which fold out inside the wall when the screw is tightened. Note they do not adust for thickness - you need to size them for the thickness of the wallboard you're using. The pictured one is probably for half to 3/4 inch depth. A 1/4 inch or 6mm thick wallboard would have less shank on it.

Not pictured are the super-cheap tube ones that come with many cheap goods like smoke detectors. Not worth it at all.

Note, once installed, none of these plugs are going to come out without leaving a large hole.

Your other option is to plate the area under the curtain rail with some solid wood as reinforcement. 10mm thick would be a minimum IMO. Screw that to the studs where they cross, and then you can hang the curtain rail hooks whereever you need them.

Number 3 is extraordinarily strong. I use them whenever possible
– vikingsteveJan 22 at 10:50

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I really love the middle anchor as well (E-Z Ancor) due to being able to install them quickly with a drill. Although they are described as "self-tapping", I prefer to give them a small pilot hole for alignment and then proceed to then drill all of the necessary anchors at once after preparing a surface.
– ZhroJan 22 at 12:48

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I have found toggle bolts to be a good option for plaster. The others can end up cracking the plaster and falling out.
– JimmyJamesJan 22 at 15:22

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Don't even think of using standard plugs (rightmost) on plasterboard. They'll pull out and damage the board when they do. They are for use in bricks, where they hold superbly well. They also work in lightweight concrete blocks, but there are special versions (none of the above) which are sold explicitly for concrete blocks and which hold much better.
– nigel222Jan 22 at 16:26

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@vikingsteve Which #3? #3 from the left or #3 from the right?
– conmanJan 22 at 19:52

As strange as this might sound, provided there's a stud back there, you can use toothpicks to fill in the hole. The toothpicks can bite into the screw, and provide friction against the surrounding wood.

If this is just drywall, use a drywall anchor like the other answers detailed

Good tip. Before those plastic expanding plugs (like in the 50's), they used to pound round wood blocks into holes and put a wood screw in there. I've seen this in old block walls in old buildings.
– element11Jan 21 at 14:43

But if there is solid wood of sufficient dimensions behind the plaster you use a long enough screw to thread into the wood. In your case, if you have good wood behind the plaster, but you have drilled too large a hole for the screw, just use a long enough anchor to fit an inch or more into the wood and thread in the screw. The anchor will expand as the screw goes in and will grip the sides of the hole in the wood.

For oversized holes in wood, lightly tap in a glue coated dowel and redrill the screw hole. This may require widening the hole to the next available dowel size. For holes in un-backed drywall, use an anchor. (Your hardware store will have various kinds, but the simple plastic ones usually work fine for light loads.) If the particular hole in question try inserting an anchor first. If it has wood behind it that prevents use of an anchor, use the dowel and cut it flush to the drywall surface. Fill in any gaps between the dowel and the drywall with spackling, drywall mud or other filler recommended by your hardware store.

Instead of a dowel you could squirt some wood glue in hole and put wooded toothpicks in until you can't fit anymore. Wait a few hours and break off the toothpicks flush with the drywall. You should be able to use the original screw. I am a finish carpenter and have done this before.

I think that using anchors, already mentioned, is the best solution. Besides the already mentioned answers, there are many different options available to you:

Drill an 8mm hole in the bracket and use a 7 mm screw into the 6mm hole.

Drill another hole in the bracket that is about a centimeter or so away from the original hole and insert a screw through that hole. Generally, the hole in the bracket should be 1 or 2 mm larger than the diameter of the screw.

If your hole is too large to take a wall plug or anchor, then your best solution is to fill the hole with filler (e.g. Polyfilla), leave it to fully harden, and then redrill the hole to the correct size. You don't need to worry too much about filling the hole perfectly flush - it'll be covered by the plate anyway.

Looking at your hole though, I suspect you may just want a larger diameter.

Note that all screws into plasterboard/drywall should use some kind of appropriate wall plug or anchor for plasterboard. Contrary to spurious claims by some screw manufacturers, it is not possible to securely fasten a screw directly into plasterboard. The point loads on the screw thread will always weaken the plasterboard, and the fixing will inevitably pull loose after not very long. This is particularly going to be an issue for your blind - you don't want it falling off!